For lots of cases, when the Tarski-Seidenberg theorem is not applicable, we can turn to the o-minimal structure which is originated from real algebraic geometry and can cover more complicated cases. We refer to [1]}, [2]} for its definition. Verifying the
o-minimal structure directly with the definition is much more complicated compared to verifying the semialgebraic set and we will focus on the existed results that can be employed directly.
Let \(\Delta \)  be an o-minimal structure. A set \(A\)  is called definable if \(A\in \Delta \) . A map \(f\)  is said to be definable if its graph \(\operatorname{gph}f \subseteq \mathbb {R}^{m+n}\)  is definable [1]}. Due to their dramatic impacts, these structures are being studied extensively. One of the interests of such structures in optimization is due to the following nonsmooth extension of KL property [2]}, [5]} (see Theorem 11 of [5]}).
